Calculating Probabilities and Expectation Values for Hydrogen Atom Wave Function

[Shafikae]

Consider a hydrogen atom whose wave function is at t=0 is the following superposition of energy eigenfunctions $$\psi$$_nlm(r)
$$\Psi$$(r, t=0) = $$\frac{1}{\sqrt{14}}$$ *[2$$\psi$$₁₀₀(r) -3$$\psi$$₂₀₀(r) +$$\psi$$₃₂₂(r)

What is the probability of finding the system in the ground state (100? in the state (200)? in the state (322)? In another energy eigenstate?
For this part i found each eigen state and put it into an integral. Should there be limits of integration for r? If so, from where to where? I did the integration for (100) and (200) but for (322) i got something crazy.

What is the expectation value of the energy: of the operator L², of the operator L_z
I have no clue what to do here.

 

[alxm]

You integrate over all of space. But to give a hint, if you recall (or read up on) the properties of eigenfunctions, you'll realize you don't need to explicitly do the integration here.

 

[Shafikae]

I know that if the subscripts are different then the int of
$$\psi$$* $$\psi$$ =0
I got something crazy for
$$\psi$$₃₂₂

but I was able to get the other 2 states. Am I suppose to integrate in terms of r from 0 to $$\infty$$ ?
Thank you.